The set S of square matrices of same order with respect to matrix addi...
Properties of the set S of square matrices:
- Quasi group:
A quasi group is a set with a binary operation that has unique solutions to both left and right division. In the set S of square matrices with matrix addition as the operation, it is not a quasi group because not all elements have unique left and right inverses.
- Semi group:
A semi group is a set with an associative binary operation. The set S of square matrices with matrix addition is closed under addition and satisfies the associative property. Therefore, it is a semi group.
- Group:
A group is a set with an associative binary operation, identity element, and inverses for each element. The set S of square matrices with matrix addition does not have inverses for each element, so it is not a group.
- Abelian group:
An abelian group is a group where the binary operation is commutative. The set S of square matrices with matrix addition is closed under addition and commutative, but lacks inverses for each element, so it is not an abelian group.
In conclusion, the set S of square matrices with respect to matrix addition is a semi group but not a group or an abelian group due to the absence of inverses for each element.